CMB Observables and Their Cosmological Implications

TL;DR

The paper demonstrates that BOOMERanG and MAXIMA CMB power-spectrum measurements can be compactly described by four observables: the first-peak position $\ell_1$ and the peak-height ratios $H_1$, $H_2$, and $H_3$, enabling a transparent link between data and cosmological parameters. Using template CDM models and external constraints, the authors derive robust flat $\Lambda$CDM limits such as $\Omega_m h^{3.8} > 0.079$, $n>0.85$, $\Omega_b h^2>0.019$, and $\Omega_m h^2<0.42$, while mapping these into an $(\Omega_m,h)$ region that favors high $h$ and moderate $\Omega_m$ (0.25–0.6). The analysis highlights the interplay between CMB geometry and peak morphology, and shows consistency with independent probes like nucleosynthesis, cluster abundance, and age bounds within an adiabatic CDM framework. The approach provides a clear, testable set of predictions for future observations and a framework to diagnose potential deviations from standard cosmology.

Abstract

We show that recent measurements of the power spectrum of cosmic microwave background anisotropies by BOOMERanG and MAXIMA can be characterized by four observables, the position of the first acoustic peak l_1= 206 pm 6, the height of the first peak relative to COBE normalization H_1= 7.6 pm 1.4, the height of the second peak relative to the first H_2 = 0.38 pm 0.04, and the height of the third peak relative to the first H_3 = 0.43 pm 0.07. This phenomenological representation of the measurements complements more detailed likelihood analyses in multidimensional parameter space, clarifying the dependence on prior assumptions and the specific aspects of the data leading to the constraints. We illustrate their use in the flat LCDM family of models, where we find Omega_m h^{3.8} > 0.079 (or nearly equivalently, the age of the universe t_0 < 13-14 Gyr) from l_1 and a baryon density Omega_b h^2 > 0.019, a matter density Omega_m h^2 < 0.42 and tilt n>0.85 from the peak heights (95% CL). With the aid of several external constraints, notably nucleosynthesis, the age of the universe and the cluster abundance and baryon fraction, we construct the allowed region in the (Omega_m,h) plane; it points to high h (0.6< h < 0.9) and moderate Omega_m (0.25 < Omega_m < 0.6).

Figures (11)

• Figure 1: Power spectrum data and models: (left panel) full range on a log scale; (right panel) first 3 peaks on a linear scale. The BOOMERanG (BOOM) and MAXIMA (MAX) points have been shifted by their 1$\sigma$ calibration errors, $10\%$ up and $4\%$ down respectively. Dashed lines represent a model that is a good fit to the CMB data alone: $\Omega_m=0.3$, $\Omega_\Lambda=0.7$, $h=0.9$, $\Omega_b h^2=0.03$, $n=1$ which gives $\ell_1= 205$, $H_1=6.6$, $H_2=0.37$, $H_3=0.52$. Solid lines represent a model that is allowed by our joint constraints: $\Omega_m=0.35$, $\Omega_\Lambda=0.65$, $h=0.75$, $\Omega_b h^2=0.023$, $n=0.95$ which gives $\ell_1= 209$, $H_1=5.8$, $H_2=0.45$, $H_3=0.5$. Note that the labelling of the $H$'s in the figure is schematic; these values are the power ratios as defined in the text.
• Figure 2: Constraints on the first peak position: $\Delta \chi^2(\ell_1)$ for the data from $75 < \ell < 375$. We define the $1 \sigma$ errors to be $1/2.5$ of the errors at $2.5 \sigma$ ($\Delta \chi^2=6.2$, solid lines).
• Figure 3: Constraint on the height of the second peak relative to the first: $\Delta \chi^2(H_2)$ for the data from $75 < \ell < 600$. $1\sigma$ errors are defined as in Fig. \ref{['fig:l1']}.
• Figure 4: Constraints on the height of the third peak relative to the first: $\Delta \chi^2 (H_3)$ for the data from $75 < \ell < 375$ and $600 < \ell < 800$. $1\sigma$ errors are defined as in Fig. \ref{['fig:l1']}.
• Figure 5: Peak position constraint in the ($\Omega_m$, $\Omega_\Lambda$) plane with various priors. The prior assumptions weaken from the light shaded region to the dark shaded region and '...' means that the unlisted priors are unchanged from the neighboring region of stronger priors. The constrained region is strongly limited by the range in $h$ considered and to a lesser extent that in $\Omega_b h^2$ and the equation of state of $\Lambda$, $w$.